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I'm told that the the power series:

[tex]\sum_{n=0} ^ \infty (2n+1) z^n[/tex] has the radius of convergens

R = 1.

Proof:

Using the Definition of convergens for power series:

[tex]\frac{(2n+1)}{(2n+1)+1} = \frac{(2n+1)}{(2n+3)} [/tex]

[tex]limit _{n \rightarrow \infty} \frac{(2n+1)}{(2n+3)} = 1[/tex]

Therefore the radius of convergens is R = 1. Right ?

Second question: The Power series above suposedly diverges on every point on the circle of convergens. How do I show that?

I know that according to the definition of divergens of the power series:

[tex]\sum_{n = 0} ^{\infty} a_n z^n [/tex]

that [tex]a_n \rightarrow \infty [/tex] for [tex]n \rightarrow \infty [/tex]

Do I the use this fact here to show that a_n diverges ??

Best Regards

Mathboy20

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# Homework Help: Radius of Convergens

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